Abstract
We prove some stability results for a simple numerical method for Mellin convolution operators with conjugation and investigate the Fredholm property for its local operators.
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References
V.D. Didenko, V.N. Matskul, On the approximative solution of singular integral equations with conjugation.Zh. Vychisl. Mat. i Mat. Fiz., 29, p. 392–404, 1989; English translation inUSSR Comput. Math. and Math. Physics 29, 1989.
V.D. Didenko, S. Roch, B. Silbermann, Approximation methods for singular integral equations on curves with corners,SIAM J. Numer. Anal., submitted.
R.V. Duduchava,Integral equations with fixed singularities, Teubner Verlagsgesellschaft, Leipzig, 1979.
R.V. Duduchava, On general singular integral operators of the plane theory of elasticity,Rend. Sem. Matem. Politec. Torino, 42, p. 15–41, 1984.
D. Elliott, A Galerkin-Petrov method for singular integral equations,J. Austral. Math. Soc. Ser. B, 25, p. 261–275, 1983.
D. Elliott, Rates of convergence for the method of classical collocation for solving singular integral equations,SIAM J. Numer. Anal., 21, p. 136–148, 1984.
D. Elliott, A comprehensive approach to the approximate solution of singular integral equations over the arc (−1,1),J. of Integral Equations and Applications, 2, p. 59–94, 1989.
J. Elschner, Asymptotics of solution to pseudodifferential equations of Mellin type,Mathem. Nachrichten, 130, p., 1987.
I.U. Gohberg, N.Ya. Krupnik, On an algebra generated by Toeplitz matrices,Funkt. Anal. u evo Pril., 3, p. 46–56, 1969.
I.C. Gohberg, I.A. Fel'dman,Convolution equations and projection methods for their solution, AMS, Translations of mathematical monographs, vol. 41, 1974.
I. Gohberg, N. Krupnik,One-dimensional linear singular integral equations, vol. 1-Operator Theory, vol. 53, Birkhäuser Verlag, Basel, 1992.
R. Hagen, S. Roch, B. Silbermann,Spectral theory of approximation methods for convolution operators, Birkhäuser Verlag, in preparation.
P. Junghanns, B. Silbermann, The numerical treatment of singular integral equations by means of polynomial approximations, preprint P-MATH-35/86, Akad. Wiss. DDR, Inst. Math., Berlin, 1986.
A.N. Kalandiya,Mathematical methods in the theory of two-dimensional elasticity, Moscow, Nauka, 1973 (in Russian).
I.K. Lifanov, Ya.E. Polonskii, The foundations of numerical methods of discrete vortices for solution of singular integral equations,PMM, 39, p. 742–746, 1975.
G.S. Litvinchuk,Boundary problems and singular integral equations with shift, Moscow, Nauka, 1977 (in Russian).
G. Monegato, S. Proessdorf, On the numerical treatment of an integral equation arising from a cruciform crack problem,Mathematical Methods in the Applied Sciences, 12, p. 489–502, 1990.
V.V. Panasiuk, M.P. Savruk, O.P. Datzishin, On the stress distribution in the neighborhood of cracks in plates and membranes,Naukova Dumka, Kiev, 1976 (in Russian).
S. Proessdorf, A. Rathsfeld, Quadrature methods for strongly elliptic Cauchy singular integral equations on an interval,Operator Theory: Advances and Applications, 41, p. 435–471, 1989.
S. Proessdorf, A. Rathsfeld, On an integral equation of the first kind arising from a cruciform crack problem, inIntegral equations and Inverse Problems, (Petrov, Sendov Editors), Longman, Coventry, p. 210–219, 1991.
S. Proessdorf, B. Silbermann,Numerical Analysis for integral and related operator equations, Birkhäuser, 1991.
A. Rathsfeld, Eine Quadraturformelmethode für Mellin-Operatoren nullter Ordnung,Mathem. Nachrichten, 137, p. 321–354, 1988.
A. Rathsfeld, A quadrature method for a Cauchy singular integral equation with fixed singularities Seminar Analysis,Oper. Equat. and Numer. Anal., 1987/88, Karl-Weierstrass-Institut, für Mathematik, Berlin, p. 107–117, 1988.
R.P. Srivastav, On the numerical solution of singular integral equations using Gauss type formulae — I: quadrature and collocation on Chebyshev nodes,IMA J. Numer. Anal. 3, p. 305–318, 1983.
E. Venturino, Stability and convergence of a hyperbolic tangent method for singular integral equations,Mathematische Nachrichten, vol. 164, p. 167–186, 1993.
E. Venturino, The Galerkin method for singular integral equations revisited,Journal of Computational and Applied Mathematics, vol. 40, p. 91–102, 1992.
F. Zhang, A hyperbolic tangent quadrature rule for solving singular integral equations with Hadamard finite part integrals,Comp. Math. Applic., 22, p. 59–73, 1991.